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erdős #391

← #390 · #392 → (packet.json; erdosproblems.com)

Let $t (n)$ be maximal such that there is a representation $n! = a_{1} \dots a_{n}$ with $t (n) = a_{1} \leq \dots \leq a_{n}$ . Obtain good bounds for $t (n) / n$ . In particular, is it true that $lim \frac{t ( n )}{n} = \frac{1}{e} ?$ Furthermore, does there exist some constant $c > 0$ such that $\frac{t ( n )}{n} \leq \frac{1}{e} - \frac{c}{lo g n}$ for infinitely many $n$ ?

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number theory · solved · 0 attempts

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A034258 — Write n! as a product of n numbers, n! = k(1)*k(2)*...*k(n) with k(1) <= k(2) <= ..., in all possible ways; a(n) = max value of k(1).1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,8,8,8,9,9,10,10,10,10,11,11,12,12,12,12,12,12,13,13,13,14,14,15,A034259 — Smallest m such that A034258(m) >= n.1,4,9,14,16,20,24,27,32,34,38,40,46,49,51,57,58,62,65,68,72,77,80,84,87,90,93,94,100,104,108,111,114,115,118,125,125,128

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finding.noted · reviewer:will-blair · 1 day

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